<s>
A	O
summed-area	B-Algorithm
table	I-Algorithm
is	O
a	O
data	B-General_Concept
structure	I-General_Concept
and	O
algorithm	O
for	O
quickly	O
and	O
efficiently	O
generating	O
the	O
sum	O
of	O
values	O
in	O
a	O
rectangular	O
subset	O
of	O
a	O
grid	O
.	O
</s>
<s>
In	O
the	O
image	B-Algorithm
processing	I-Algorithm
domain	O
,	O
it	O
is	O
also	O
known	O
as	O
an	O
integral	B-Algorithm
image	I-Algorithm
.	O
</s>
<s>
It	O
was	O
introduced	O
to	O
computer	O
graphics	O
in	O
1984	O
by	O
Frank	O
Crow	O
for	O
use	O
with	O
mipmaps	B-Data_Structure
.	O
</s>
<s>
In	O
computer	B-Application
vision	I-Application
it	O
was	O
popularized	O
by	O
Lewis	O
and	O
then	O
given	O
the	O
name	O
"	O
integral	B-Algorithm
image	I-Algorithm
"	O
and	O
prominently	O
used	O
within	O
the	O
Viola	B-General_Concept
–	I-General_Concept
Jones	I-General_Concept
object	I-General_Concept
detection	I-General_Concept
framework	I-General_Concept
in	O
2001	O
.	O
</s>
<s>
As	O
the	O
name	O
suggests	O
,	O
the	O
value	O
at	O
any	O
point	O
(	O
x	O
,	O
y	O
)	O
in	O
the	O
summed-area	B-Algorithm
table	I-Algorithm
is	O
the	O
sum	O
of	O
all	O
the	O
pixels	O
above	O
and	O
to	O
the	O
left	O
of	O
(	O
x	O
,	O
y	O
)	O
,	O
inclusive	O
:	O
</s>
<s>
The	O
summed-area	B-Algorithm
table	I-Algorithm
can	O
be	O
computed	O
efficiently	O
in	O
a	O
single	O
pass	O
over	O
the	O
image	O
,	O
as	O
the	O
value	O
in	O
the	O
summed-area	B-Algorithm
table	I-Algorithm
at	O
(	O
x	O
,	O
y	O
)	O
is	O
just	O
:	O
</s>
<s>
Once	O
the	O
summed-area	B-Algorithm
table	I-Algorithm
has	O
been	O
computed	O
,	O
evaluating	O
the	O
sum	O
of	O
intensities	O
over	O
any	O
rectangular	O
area	O
requires	O
exactly	O
four	O
array	O
references	O
regardless	O
of	O
the	O
area	O
size	O
.	O
</s>
<s>
where	O
is	O
the	O
integral	B-Algorithm
image	I-Algorithm
at	O
and	O
the	O
image	O
dimension	O
.	O
</s>
<s>
In	O
neuroimaging	B-Algorithm
,	O
for	O
example	O
,	O
the	O
images	O
have	O
dimension	O
or	O
,	O
when	O
using	O
voxels	B-Algorithm
or	O
voxels	B-Algorithm
with	O
a	O
time-stamp	O
.	O
</s>
<s>
This	O
method	O
has	O
been	O
extended	O
to	O
high-order	O
integral	B-Algorithm
image	I-Algorithm
as	O
in	O
the	O
work	O
of	O
Phan	O
et	O
al	O
.	O
</s>
<s>
who	O
provided	O
two	O
,	O
three	O
,	O
or	O
four	O
integral	B-Algorithm
images	I-Algorithm
for	O
quickly	O
and	O
efficiently	O
calculating	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
(	O
variance	O
)	O
,	O
skewness	O
,	O
and	O
kurtosis	O
of	O
local	O
block	O
in	O
the	O
image	O
.	O
</s>
<s>
To	O
compute	O
variance	O
or	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
a	O
block	O
,	O
we	O
need	O
two	O
integral	B-Algorithm
images	I-Algorithm
:	O
</s>
<s>
and	O
are	O
computed	O
quickly	O
by	O
integral	B-Algorithm
image	I-Algorithm
.	O
</s>
<s>
Similar	O
to	O
the	O
estimation	O
of	O
the	O
mean	O
(	O
)	O
and	O
variance	O
(	O
)	O
,	O
which	O
requires	O
the	O
integral	B-Algorithm
images	I-Algorithm
of	O
the	O
first	O
and	O
second	O
power	O
of	O
the	O
image	O
respectively	O
(	O
i.e.	O
</s>
<s>
is	O
that	O
of	O
integer	O
overflow	O
occurring	O
for	O
the	O
higher	O
order	O
integral	B-Algorithm
images	I-Algorithm
in	O
case	O
32-bit	O
integers	O
are	O
used	O
.	O
</s>
